APSTA-GE 2003: Intermediate Quantitative Methods

class: center, middle, inverse, title-slide

# APSTA-GE 2003: Intermediate Quantitative Methods
## Lab Section 003, Week 6
### New York University
### 10/13/2020

---

## Reminders

- **Group Assignment**

- Due: **10/14/2020 11:59pm (EST)**
  
- **Assignment 3**

- Due: **10/19/2020 11:55pm (EST)**
  
- Office hours

- Monday 9 - 10am (EST)
  - Wednesday 12:30 - 1:30pm (EST)
  - By Appointments
  
- Office hour Zoom link
  - https://nyu.zoom.us/j/97347070628 (pin: 2003)
  
- Office hour notes
  - Available on NYU Classes
- Updates on Lab Slides
  - Available on NYU Classes

---

## Today's Topics

- Review Diagnostic Regression Analysis

- How to check **linearity**?
  
      - Residual vs. Fitted Values
    
  - How to check the **normality** of residuals?
  
      - QQ-plot
      
  - How to check **equal variance**?
  
      - Residual vs. Fitted Values
      
  - How to find outliers?
  
      - Outliers
      
      - Leverage Points
      
      - Influential Points
    
- Regression Diagnosis

- Using a new dataset: `marketing`

---
class: inverse, center, middle

# Diagnostic Regression Analysis

---

## Linear Regression

### **Linear + Regression**

### An approach to:

1. Make prediction by fitting a regression model.

2. Explain how much variance of the dependent variable can be explained by independent variable(s).

**Simple linear regression:** with only one independent variable

**Multiple linear regression:** contains two or more independent variables

---

## Model Types

- **Linear equation**: `$Y_i = \beta_1 + \beta_2 \times X_i$`
- **Log equation:** `$Y_i = \beta_1 + \beta_2 \times log(X_i)$`
- **Quadratic equation:** `$Y_i = \beta_1 + \beta_2 \times X_i^2$`
- **Cubic equation:** `$Y_i = \beta_1 + \beta_2 \times X_i^3$`

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---

## Model Types Visualization

- Linear model: `$Y_i = 20 + 3 \times X_i$`
- Log model: `$Y_i = 20 + 3 \times log(X_i)$`
- Quadratic model: `$Y_i = 20 + 3 \times X_i^2$`
- Cubic model `$Y_i = 20 + 3 \times X_i^3$`
 - `$\varepsilon \sim \mathcal{N}(0, 1)$`

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---

## How to check linearity?

What type of model should we use? （☉∀☉）

```r
n <- 50 # Sample size
X <- rnorm(n, mean = 3, sd = 1) # I.V.
dat_tricky <- data.frame(
 X, 
* Y = 10 + 2 * X + 15 * log(X) -5 * X^2 + X^3 + rnorm(n, mean = 0, sd = 20)
)
```

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---

## To Check Linearity...

Let's fit a linear regression model using the correct dataset first!

```r
str(dat_linear)
```

```
## 'data.frame':	50 obs. of  2 variables:
##  $ X: num  2.91 4.32 3.64 4.17 3.12 ...
##  $ Y: num  28 32.4 30.6 31.2 29.9 ...
```

```r
# Fit a linear regression model
*mod_lin <- lm(Y ~ X, data = dat_linear)
*summary(mod_lin)
```

```
## 
## Call:
## lm(formula = Y ~ X, data = dat_linear)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1.7693 -0.7461 -0.2736 0.5419 2.6855 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 20.6092 0.4265 48.32 <2e-16 ***
## X 2.8983 0.1252 23.14 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.035 on 48 degrees of freedom
## Multiple R-squared: 0.9178,	Adjusted R-squared: 0.916 
## F-statistic: 535.6 on 1 and 48 DF, p-value: < 2.2e-16
```

---

## After Fitting the Model...

We can check model linearity by visualizing the distribution of residuals.

```r
par(mfrow = c(1, 2))
*plot(Y ~ X, data = dat_linear)
text(Y ~ X, data = dat_linear, labels = rownames(dat_linear), cex = 0.6, pos = 3)
abline(a = 19.8578, b = 3.0524, col = "red")
*plot(mod_lin, which = 1)
```

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/check-linear-2-1.png)

---

## Res. vs. Fitted for Log Model

```r
*mod_log <- lm(Y ~ log(X), data = dat_log)
par(mfrow = c(1, 2))
*plot(Y ~ log(X), data = dat_log)
text(Y ~ log(X), data = dat_log, labels = rownames(dat_log), cex = 0.6, pos = 3)
*plot(mod_log, which = 1)
```

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/demo-log-viz-1.png)

---

## Res. vs. Fitted for Quadratic Model

```r
*mod_quad <- lm(Y ~ X^2, data = dat_quad)
par(mfrow = c(1, 2))
*plot(Y ~ X^2, data = dat_quad)
text(Y ~ X^2, data = dat_quad, labels = rownames(dat_quad), cex = 0.6, pos = 3)
*plot(mod_quad, which = 1)
```

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/demo-quad-viz-1.png)

---

## Res. vs. Fitted for Cubic Model

```r
*mod_cubic <- lm(Y ~ X^3, data = dat_cubic)
par(mfrow = c(1, 2))
*plot(Y ~ X^3, data = dat_cubic)
text(Y ~ X^3, data = dat_cubic, labels = rownames(dat_cubic), cex = 0.6, pos = 3)
*plot(mod_cubic, which = 1)
```

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/demo-cubic-viz-1.png)

---

## Residuals vs. Fitted for Different Models

```r
par(mfrow = c(2, 2))
*mod_lin <- lm(Y ~ X, data = dat_linear)
*plot(lm(Y ~ X, data = dat_linear), which = 1)
*plot(lm(Y ~ log(X), data = dat_log), which = 1)
*plot(lm(Y ~ X^2, data = dat_quad), which = 1)
*plot(lm(Y ~ X^3, data = dat_cubic), which = 1)
```

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/demo-all-viz-1.png)

---

## How to check the normality of residuals?

Given a linear regression model, we can check the **normality** of residuals using a **Q-Q plot**.

```r
par(mfrow = c(1, 2))
*plot(mod_lin, which = 2, pch = 16)
X_perfect = c(1: 50)
dat_perfect <- data.frame(X = X_perfect, Y = X_perfect)
plot(lm(Y ~ X, dat = dat_perfect), which = 2, pch = 16, col = "red")
```

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/qq-plot-1.png)

---

## Q-Q Plot Also Applies to Other Model Types

However, normality does not hold on certain model types.

```r
par(mfrow = c(1, 3))
*plot(mod_log, which = 2, pch = 16)
*plot(mod_quad, which = 2, pch = 16)
*plot(mod_cubic, which = 2, pch = 16)
```

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/demo-qq-viz-1.png)

---

## How to check equal variance?

To check equal variance, we can again use the **Residuals vs. Fitted** plot.

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/check-equal-var-1.png)

---

## How to find outliers?

To check outliers, we can calculate **standardized residuals** and plot against the fitted values.

```r
*library(MASS)
*dat_linear$yhat <- predict(mod_lin)
*dat_linear$sd_res <- stdres(mod_lin)
*plot(sd_res ~ yhat, data = dat_linear, ylim = c(-3, 3))
text(sd_res ~ yhat, data = dat_linear, labels = rownames(dat_linear), cex = 0.8, pos = 3)
*abline(h = 2, col = "red")
*abline(h = -2, col = "red")
```

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/stu-res-1.png)

---

## Leverage Points

Magnets. Attract vertically. Push down or pull up the regression line.

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/leverage-1.png)

---

## Influential Points

Points that have influence on both directions.

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/influence-1.png)

---

## Detecting Influential Points

```r
*library(car)
influencePlot(mod_lin)
```

![](data:image/png;base64,#Lab6-Slides-101320_files/figure-html/demo-detection-1.png)

```
##       StudRes        Hat      CookD
## 6  -0.6110870 0.16342545 0.03695707
## 31 -0.5627195 0.10545620 0.01893440
## 47  2.1316996 0.04867638 0.10826146
## 50  2.8018499 0.02002342 0.07018497
```

---

class: inverse, center, middle

# Regression Diagnosis

---

## New Dataset for Practice

Dataset: [marketing.csv](https://drive.google.com/file/d/1RO3M0__Tnb_MInU9Modj8TIIuZqZG_-p/view?usp=sharing)

```r
dat <- datarium::marketing
```

**Let's move to RStudio.**

---

## Contact

Tong Jin

- Email: tj1061@nyu.edu
- Office Hours
  - Mondays, 9 - 10am (EST)
  - Wednesdays, 12:30 - 1:30pm (EST)